Question 1:
Express the following as single fractions.
Give your answers in their simplest forms.
1(a) \dfrac{x-10}{2}+\dfrac{3x}{10}
ANSWER: Multiple Choice (Type 1)
A: \dfrac{4x-15}{5}
B: \dfrac{4x-20}{5}
C: \dfrac{4x-25}{5}
D: \dfrac{4x-30}{5}
Answer: C
Workings:
\dfrac{x-10}{2}+\dfrac{3x}{10}=\dfrac{5(x-10)}{10}+\dfrac{3x}{10}=\dfrac{8x-50}{10}=\dfrac{4x-25}{5}
Marks = 1
1(b) \dfrac{b-2}{3}-\dfrac{b}{2}
ANSWER: Multiple Choice (Type 1)
A: \dfrac{-b-4}{6}
B: \dfrac{-b-5}{6}
C: \dfrac{-b-7}{6}
D: \dfrac{-b-8}{6}
Answer: A
Workings:
\dfrac{b-2}{3}-\dfrac{b}{2}=\dfrac{2(b-2)}{6}-\dfrac{3b}{6}=\dfrac{-b-4}{6}
Marks = 1
1(c) \dfrac{a+3}{2}+\dfrac{2a-1}{3}
ANSWER: Multiple Choice (Type 1)
A: \dfrac{7(a+4)}{6}
B: \dfrac{7(a+3)}{6}
C: \dfrac{7(a+2)}{6}
D: \dfrac{7(a+1)}{6}
Answer: D
Workings:
\dfrac{a+3}{2}+\dfrac{2a-1}{3}=\dfrac{3a+9}{6}+\dfrac{4a-2}{6}=\dfrac{7a+7}{6}
=\dfrac{7(a+1)}{6}
Marks = 2
Question 2:
Express the following as single fractions.
Give your answers in their simplest forms.
2(a) \dfrac{3x+1}{6}+\dfrac{2x-2}{4}
ANSWER: Multiple Choice (Type 1)
A: \dfrac{2x-1}{3}
B: \dfrac{3x-1}{3}
C: \dfrac{4x-1}{3}
D: \dfrac{5x-1}{3}
Answer: B
Workings:
\dfrac{3x+1}{6}+\dfrac{2x-2}{4}=\dfrac{2(3x+1)}{12}+\dfrac{3(2x-2)}{12}
=\dfrac{12x-4}{12}=\dfrac{3x-1}{3}
Marks = 2
2(b) \dfrac{4}{2x-2}+\dfrac{10}{x-1}
ANSWER: Multiple Choice (Type 1)
A: \dfrac{10}{x-1}
B: \dfrac{11}{x-1}
C: \dfrac{12}{x-1}
D: \dfrac{13}{x-1}
Answer: C
Workings:
\dfrac{4}{2x-2}+\dfrac{10}{x-1}=\dfrac{4}{2x-2}+\dfrac{20}{2(x-1)}
=\dfrac{24}{2x-2}=\dfrac{12}{x-1}
Marks = 2
2(c) \dfrac{3(x+1)}{5x}+\dfrac{2x}{2x+2}
ANSWER: Multiple Choice (Type 1)
A: \dfrac{5x^2+6x+3}{2x^2+5x}
B: \dfrac{6x^2+6x+3}{3x^2+5x}
C: \dfrac{7x^2+6x+3}{4x^2+5x}
D: \dfrac{8x^2+6x+3}{5x^2+5x}
Answer: D
Workings:
\dfrac{3(x+1)}{5x}+\dfrac{x}{x+1}=\dfrac{3(x+1)(x+1)}{5x(x+1)}=\dfrac{x(5x)}{5x(x+1)}
=\dfrac{3(x^2+3x+1)+5x^2}{5x^2+5x}
=\dfrac{8x^2+6x+3}{5x^2+5x}
Marks = 3
Question 3:
Simplify fully,
3(a) \dfrac{4x^2+16x}{x^2-16}
ANSWER: Multiple Choice (Type 1)
A: \dfrac{4x}{x-4}
B: \dfrac{5x}{x-4}
C: \dfrac{6x}{x-4}
D: \dfrac{7x}{x-4}
Answer: A
Workings:
\dfrac{4x(x+4)}{(x-4)(x+4)}
=\dfrac{4x}{x-4}
Marks = 2
3(b) \dfrac{2x+4}{x+1} \div \dfrac{x^2+x-2}{2x^2+5x+3}
ANSWER: Multiple Choice (Type 1)
A: \dfrac{2x+6}{x-1}
B: \dfrac{3x+6}{x-1}
C: \dfrac{4x+6}{x-1}
D: \dfrac{5x+6}{x-1}
Answer: C
Workings:
\dfrac{2(x+2)}{x+1} \div \dfrac{(x+2)(x-1)}{(2x+3)(x+1)}
=\dfrac{2(x+2)}{x+1} \times \dfrac{(2x+3)(x+1)}{(x+2)(x-1)}
=\dfrac{2(2x+3)}{x-1}=\dfrac{4x+6}{x-1}
Marks = 3
Question 4:
Which of the following expressions is equal to \dfrac{1}{x+4}?
ANSWER: Multiple Choice (Type 1)
A: \dfrac{x-2}{x^2-6x+8}
B: \dfrac{x+4}{x^2+5x+6}
C: \dfrac{x-4}{x^2+6}
D: \dfrac{x+2}{x^2+6x+8}
Answer: D
Workings:
\dfrac{x+2}{x^2+6x+8}=\dfrac{x+2}{(x+2)(x+4)}
=\dfrac{x+2}{x^2+6x+8}
Marks = 2
Question 5
Simplify fully,
5(a) \dfrac{x-5}{x^2-25}
ANSWER: Multiple Choice (Type 1)
A: \dfrac{1}{x+5}
B: \dfrac{1}{x+6}
C: \dfrac{1}{x+7}
D: \dfrac{1}{x+8}
Answer: A
Workings:
\dfrac{x-5}{x^2-25}=\dfrac{x-5}{(x+5)(x-5)}=\dfrac{1}{x+5}
Marks = 1
5(b) \dfrac{x^2+x-6}{x^2-x-12}
ANSWER: Multiple Choice (Type 1)
A:\dfrac{x-5}{x-7}
B:\dfrac{x-4}{x-6}
C:\dfrac{x-3}{x-5}
D:\dfrac{x-2}{x-4}
Answer: D
Workings:
\dfrac{(x-2)(x+3)}{(x-4)(x+3)}
=\dfrac{x-2}{x-4}
Marks = 2
Question 6
Solve \dfrac{1}{2x-3}+\dfrac{4}{x+1}=1
ANSWER: Multiple Answers (Type 2)
Answer: x=1 or x=4
Workings:
\dfrac{(x+1)}{(2x-3)(x+1)}+\dfrac{4(2x-3)}{(2x-3)(x+1)}=1
\dfrac{x+1+8x-12}{(2x-3)(x+1)}=1
9x-11=(2x-3)(x+1)
9x-11=2x^2-x-3
0=2x^2-10x+8
x^2-5x+4=0
(x-4)(x-1)=0
x=1 or x=4
Marks = 4
Question 7
The diagram below shows a rectangle.
The perimeter of the rectangle is equal to 3 times the area.
Find the value of x.
ANSWER: Simple text answer
Answer: 2
Workings:
3ab=2a+2b
Thus,
\begin{aligned} 3\bigg[\bigg(\dfrac{2}{x+3}\bigg)\bigg(\dfrac{1}{x-3}\bigg)\bigg]&=2\bigg(\dfrac{2}{x+3}\bigg)+2\bigg(\dfrac{1}{x-3}\bigg) \\[2em] 3\bigg(\dfrac{2}{x^2-9}\bigg) &= \dfrac{4}{x+3}+\dfrac{2}{x-3} \\[2em] \dfrac{6}{x^2-9} &=\dfrac{4(x-3)+2(x+3)}{(x+3)(x-3)} \\[2em] \dfrac{6}{x^2-9} &= \dfrac{4x-12+2x+6}{x^2-9} \\[2em] \dfrac{6}{x^2-9} &=\dfrac{6x-6}{x^2-9} \\[2em] \dfrac{6x-6}{6} &=\dfrac{x^2-9}{x^2-9} \\[2em] x-1 &= 1 \\[2em] x &=2\end{aligned}Marks = 5